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| Mirrors > Home > MPE Home > Th. List > 9t11e99 | Structured version Visualization version GIF version | ||
| Description: 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| 9t11e99 | ⊢ (9 · ;11) = ;99 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 9cn 12367 | . . . 4 ⊢ 9 ∈ ℂ | |
| 2 | 10nn0 12753 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
| 3 | 2 | nn0cni 12540 | . . . . 5 ⊢ ;10 ∈ ℂ | 
| 4 | ax-1cn 11214 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 5 | 3, 4 | mulcli 11269 | . . . 4 ⊢ (;10 · 1) ∈ ℂ | 
| 6 | 1, 5, 4 | adddii 11274 | . . 3 ⊢ (9 · ((;10 · 1) + 1)) = ((9 · (;10 · 1)) + (9 · 1)) | 
| 7 | 3 | mulridi 11266 | . . . . . 6 ⊢ (;10 · 1) = ;10 | 
| 8 | 7 | oveq2i 7443 | . . . . 5 ⊢ (9 · (;10 · 1)) = (9 · ;10) | 
| 9 | 1, 3 | mulcomi 11270 | . . . . 5 ⊢ (9 · ;10) = (;10 · 9) | 
| 10 | 8, 9 | eqtri 2764 | . . . 4 ⊢ (9 · (;10 · 1)) = (;10 · 9) | 
| 11 | 1 | mulridi 11266 | . . . 4 ⊢ (9 · 1) = 9 | 
| 12 | 10, 11 | oveq12i 7444 | . . 3 ⊢ ((9 · (;10 · 1)) + (9 · 1)) = ((;10 · 9) + 9) | 
| 13 | 6, 12 | eqtri 2764 | . 2 ⊢ (9 · ((;10 · 1) + 1)) = ((;10 · 9) + 9) | 
| 14 | dfdec10 12738 | . . 3 ⊢ ;11 = ((;10 · 1) + 1) | |
| 15 | 14 | oveq2i 7443 | . 2 ⊢ (9 · ;11) = (9 · ((;10 · 1) + 1)) | 
| 16 | dfdec10 12738 | . 2 ⊢ ;99 = ((;10 · 9) + 9) | |
| 17 | 13, 15, 16 | 3eqtr4i 2774 | 1 ⊢ (9 · ;11) = ;99 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 9c9 12329 ;cdc 12735 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-dec 12736 | 
| This theorem is referenced by: 3dvds2dec 16371 1259lem3 17171 | 
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